The explicit form of the inelastic X-ray scattering, IXS, cross-section is derived within a time-dependent perturbative treatment of the scattering process. In this derivation, the double differential cross-section is obtained from the Fermi Golden Rule within a plane wave expansion of the vector potential. Furthermore, it is assumed throughout that the Thompson term of the perturbative Hamiltonian yields the overwhelming contribution to the scattering. The achievement of an explicit form for the double differential scattering cross-section rests on the validity of the adiabatic or Born-Oppenheimer approximation. As a result, it is here shown that that the IXS double differential cross-section is proportional to the spectrum of density fluctuations of the sample, which is thus the sample variable directly accessed by IXS measurements. Although the whole treatment is valid for monatomic systems only, under suitable approximations, it can be extended to molecular systems.
- inelastic X-ray scattering
- theory of the scattering
- theory of the line-shape
- double differential scattering cross-section
Inelastic scattering measurements are among the most powerful tools to investigate the collective terahertz dynamics of disordered systems [1, 2]. Although this subject has been the focus of intense scrutiny in the past few decades, it still presents many challenging aspects. In a spectroscopic measurement, the dynamic response of the target system is stimulated via the exchange of an energy ℏω and momentum ℏ
Although the spectral profile is exactly known analytically in both hydrodynamic and single-particle limits, its evolution at the crossover in between them still eludes a firm understanding. Particularly insightful appears the study of the line-shape in the mesoscopic range, which corresponds to 2π/
This range is the natural domain of high-resolution inelastic scattering, IXS , a spectroscopic method, which, since its development towards the end of the past millennium, has substantially improved the current understanding of the terahertz dynamics of condensed matter systems. This success partly owes to both inherent and practical advantages that this technique offers compared to the complementary terahertz spectroscopy, inelastic neutron scattering, INS. Intrinsic benefits include the virtual absence of kinematic limitations, the straightforward implementation of constant-
As an introduction to the field, this chapter is devoted to a derivation of the cross-section of IXS measurements, thus elucidating its direct connection with the Fourier transform of the atomic density fluctuations autocorrelation function. A similar treatment, which can also be found in Refs. [5, 8], is strictly valid for monatomic systems only, even though it can be easily generalized to the case of molecular systems.
2. Generalities on an inelastic scattering measurement
In a typical IXS measurement, a beam of particles–waves, as, for example, neutrons, X-rays or electrons, having well-defined energy, wavevector and polarization impinges on a sample and, after the impact, it is scattered all over the solid angle. A detector placed at a distance
At a long distance from the centre of the scattering, the electromagnetic wave generated by the scattering event is the sum of a plane and a spherical wave , that is, waves having respectively a planar and a spherical wavefront. In other terms, the ultimate effect of the scattering source is to remove a part of the photons from their initial ‘plane wave’ state and reradiate them into a spherical wave, which is consistent with well-known Huygens-Fresnel principle . However, it is always safe to assume that the detector—which intercepts photons deviated by an angle 2
As it appears from Figure 1, if one considers the plane defined by the two vectors and , only one angular coordinate, the scattering angle , is sufficient to describe the scattering problem.
To derive an expression of the intensity detected in high-resolution
For the sake of simplicity, a few more assumptions are here considered: (1) the sample has a straightforward shape: a slab of thickness
Under these assumptions, we can write a general expression to estimate the number of photons per unit time impinging on the detector, which is given by:
where is the photon flux on the sample, defined as the number of photons impinging on the sample per unit time and unit area, while is the number of scattering units per unit volume, which is here assumed constant throughout the X-ray-illuminated sample.
The above formula introduces the double differential scattering cross-section:
which is the only parameter of Eq. (1) conveying non-trivial information on the sample properties.
It can be recognized that the beam intensity across the sample thickness is not constant, as a part of it gets absorbed by the sample itself. This intensity reduction can be easily evaluated by expressing the attenuation caused by an elemental sample slice of thickness
where is the absorption coefficient at the energy of the incident beam. The integration of both members of the above equation leads to the conclusion that the intensity transmitted through the sample experiences an exponential decay. Assuming a forward scattering geometry, the attenuation factor can be simply obtained as and inserted in Eq. (1), thus obtaining:
The above formula lends itself to a direct estimate of the ideal sample thickness, which is identified by the condition, which yields . In summary, the optimal sample thickness should match the absorption length of the sample at the energy of the incident beam. For typical incident beam energies of most current
3. The interaction between impinging electromagnetic field and target electrons
Given this preliminary discussion, the focus is now on the analytical derivation of the
where and are the position and the momentum of the
where the unperturbed Hamiltonian, associated with the multielectron system in the absence of the electromagnetic field, reads as:
plus the other two terms accounting for the perturbation induced by the impinging electromagnetic field, that is, respectively:
and the so-called Thomson scattering term:
Here the symbol denotes the anticommutator operator, while is the classical electron radius expressed in cgs units. To its leading order, the perturbation in Eq. (8) describes one-photon interactions with the sample as absorption and emission, while two-photon processes, such as the scattering event, come into play to the second-order only. Conversely, the Thomson term (Eq. (9)), being quadratic in the vector potential, accounts to the first order for two photons interactions such as the scattering event. Away from an energy resonance, the latter term largely exceeds the second-order expansion of Eq. (8), thus providing an overwhelming contribution to the scattering process, which will be hereafter assumed to be entirely described by the Thomson term.
As mentioned, in a typical scattering measurement, the X-ray photons undergo a transition between two different plane wave states. Therefore, one could, in principle, use the Fermi Golden Rule  to count all scattered photons emanating from a single incident plane wave and having wavevector pointing to a direction to within a solid angle , thus deriving the double differential cross-section explicitly.
This strategy would require, in principle, a proper normalization of the photon wave functions, but, unfortunately, plane waves have normalization integral diverging for long distances. This difficulty is usually circumvented by confining the description of the scattering problem to a cubic box of size
Here the indexes ‘
Coming back to the double differential scattering cross-section, one can express it as:
where is the probability rate per sample and probe units that a photon experiences a transition between the initial and the final photon states, while the term represents the density of final photon states. The probability rate in Eq. (11) should be more appropriately written as a sum over all elementary excitations in the sample possibly coupling with the scattering event. Hence,
with denoting the probability of a transition between the combined states of the photon and the sample, labeled by lower case and capital fonts, respectively.
Eq. (12) is particularly useful as the term under summation can be derived explicitly using the Fermi Golden Rule, according to which:
The last factor in the right-hand side of the above equation contains the perturbative part of the Hamiltonian computed between initial and final combined photon and sample states. As mentioned, we will assume that this term entirely coincides with the Thomson term in Eq. (9).
At this stage, the derivation of the double differential cross-section requires one to tackle the density of final states analytically, as discussed in the next paragraph.
4. Counting the photon states
It is worth noticing that the expedient of circumscribing the scattering within a
In the reciprocal space, the bandwidth corresponds to the volume of the spherical shell of infinitesimal thickness represented in Figure 2, for which one can write . The wavevectors’ components in the box
where , and are generic integers.
The set of wavevectors defined in Eq. (15) identifies a lattice in
Therefore, one has
For photons, the link between energy and wavevector is fixed by the linear law , which can be differentiated to obtain , thus eventually getting
which, combined with Eq. (11), yields
At this stage, the interaction term, that is, the squared matrix element appearing in the Fermi Golden Rule (Eq. (13)), can be made explicit by inserting in it the Thomson term in Eq. (9), while using the expression of the vector potential in Eq. (10), thus eventually obtaining
where the vector is the position of the
Here, is the energy gained by the photons in the scattering process, while the
5. From the adiabatic approximation to the dynamic structure factor
The right-hand side of Eq. (21) contains three independent factors, the integral term being the only one directly relating to the properties of the target sample. The latter can hardly be handled analytically, due to the complex interplay between electrons belonging to different atoms, which couples electronic and nuclear coordinates. However, it becomes treatable under the reasonable approximation that the centre of mass of the electronic cloud drifts following with no delay the slow nuclear motion. This assumption is customarily referred to as ‘adiabatic’, or Born-Oppenheimer, approximation ; its use justifies the factorization of the target system ‘ket’ as , with nuclear and electronic states being labeled by the suffixes ‘
Within the validity of these assumptions, the double differential cross-section in Eq. (18) reduces to
where and are the energies associated with the initial and final nuclear states respectively, and
is the form factor of the
If a single atomic species is present in the sample, all atoms have the same form factor, that is, ; this further simplifies the expression of the double differential cross-section
The above expression can be cast in a more compact form after the few additional manipulations as the use of an integral representation of the δ-function of energy, the Heisenberg representation of a time-dependent operator and the completeness of the final eigenstate.
In its initial state, the sample is usually a many-atoms system at equilibrium, and the sum over its initial state can be computed as an ordinary equilibrium average, which leads to the following identity:
where as usual, the angle brackets denote the thermal average on the system at equilibrium. The expression above is the time correlation function of the variable , which involves the pair composed by the
In summary, as a result of all manipulations mentioned above, the double differential cross-section in Eq. (24) eventually reduces to
6. Introducing a key stochastic variable: the microscopic density fluctuation
The expression between angle brackets is the equilibrium autocorrelation function of the dynamic variable , which involves the positions of the generic
The interpretation of this function as a microscopic density is perhaps more evident as one considers its average value over the whole sample volume:
which is consistent with the macroscopic definition of number density. Notice that the δ-function is an extremely irregular discontinuous profile, which however adequately accounts for the atomistic, character of the system.
In the reciprocal space, one deals with the Fourier transform of the microscopic density, namely
Furthermore, since scattering phenomena arise from inhomogeneities or fluctuation from equilibrium, we are here mainly interested in the microscopic density fluctuation:
Again, the variable of direct pertinence for a spectroscopic measurement is instead the Fourier transform of such a fluctuation:
in which it was considered that the Fourier transform of a constant function is a δ-function.
We can now introduce the intermediate scattering function as the space Fourier transform of the correlation function between density fluctuations:
and its one-sided time Fourier transform
which is customarily referred to as the spectrum of density fluctuations, or the dynamic structure factor of the system.
7. The double differential cross-section and the dynamic structure factor
Given the dynamic variables introduced in the previous section, it can be readily verified that Eq. (26) can be cast in the more compact form:
is the spectrum associated with the dynamic variable. Notice that for a homogeneous and isotropic system such as a liquid, such a variable does not depend on the direction of the exchanged wavevector, but uniquely on its amplitude .
Let us discuss here how the spectrum in Eq. (35) relates to the variable density fluctuations as defined by Eq. (30). By definition, the spectrum of such a variable is the Fourier transform of the autocorrelation function. Explicitly
with . At this stage, one can define the spectrum of the microscopic density as
It appears that the spectra of either or , which are labeled by the respective indexes and , differ by a term proportional to the product .
This term accounts for the forward transmitted elastic scattering, which is of no relevance for a scattering experiment as it describes the signal from photons that have exchanged no energy or momentum with the target sample.
In practice, such a signal is never detected by scattering measurements, as it does not convey insight into non-trivial samples properties; furthermore, it fully overlaps with the forward transmitted beam, which is often so intense to burn or damage detectors. For these reasons, IXS measurements are always performed at finite scattering angles, where one has
As discussed, the identity above entails the replacement of the microscopic density with its fluctuation from equilibrium
This expression of the cross-section above has been derived assuming a target sample composed by
A more detailed treatment of this problem within the hypothesis of random molecular orientations and weak coupling between orientational and translational degrees of freedom leads to the conclusion that the spectrum splits into a coherent and an incoherent component. Consequently, the cross-section can be cast in the following general form:
where , where the suffix ‘’ indicates an average over molecular orientations, while the suffixes ‘I’ and ‘C’ label the incoherent and coherent parts of the dynamic structure factor.
8. An estimate of the count rate
An estimate of the count rate achievable by an IXS measurement can be worked out starting from the expression of the total scattering cross-section, while assuming, for instance, a sample having an optimal thickness . The flux of scattered photons in the solid angle and the energy interval is thus given by
At this stage, both members of the equation can be integrated in time, and the double integration over both solid angle and final energy must also be performed to obtain the total cross-section of the IXS scattering. In the low limit, where the atomic form factor and that the approximation , one has:
where , while is the absorption cross-section. An idea of the counting efficiency of IXS is provided by Figure 3, which displays the value of the ratio for an incident X-ray beam having 10 keV energy, as a function of the atomic number. The abrupt increase of this parameter can be readily appreciated at the absorption above the K-edge, that is, above the binding energy of the innermost electron shell; these innermost electrons are those primarily interacting with the incident X-ray.
8.1 The signal measured by a real instrument
As a result of the previous treatment, it was demonstrated that the cross-section is proportional to . However, such a treatment is entirely classical, insofar as all relevant observables are treated as commuting variables. Quantum effects are accounted for only through the so-called detailed balance principle, which takes into account the statistical population of the various -states of the sample. These effects ultimately result in an asymmetry of the spectrum respect to its elastic, , position. The most popular recipe for handling them is to assume that the true spectrum can be obtained from the classic, symmetric, counterpart by adding a suitable frequency-dependent factor. Explicitly,
Still, the above formula does not capture two essential aspects of the measured scattering signal, as the contribution of the instrumental resolution and the spectral background. These are explicitly accounted for by using the following general expression for the intensity profile:
where is an overall intensity factor, while the usually mildly frequency-dependent coefficient accounts in principle for both the spectral background and the dark counts of a detector; when modeling the line-shape, sometimes the latter coefficient is assumed either constant or linearly dependent on .
The resolution profile represents the instrumental rendering of a spectral shape having zero energy width, that is, the δ(ω)–profile representing a perfectly elastic scattering. In a typical IXS measurement, such a resolution is estimated by measuring the scattering signal from an almost perfect elastic scatterer, often identified in a sample of plexiglas at the
8.2 A practical example: a comparison between an IXS and an INS measurement
In general, the kinematic laws ruling the scattering process impose some limitations to the dynamic Q,ω region explorable by the measurement. These kinematic constraints are especially severe for inelastic neutron scattering, INS . Although these limitations are irrelevant for IXS, the portion of the dynamic plane explored by this technique is still limited in the low-energy, or low-frequency, side by the finite instrumental energy resolution.
Figure 4 provides a clear example of how resolution and kinematic limitations differently affect IXS and INS. Indeed, the plot compares the spectra measured in a joint INS and IXS measurement on the same sample of heavy water , after normalization of the respective areas. The elastic peak in the INS spectrum has a spike-like shape. Such a sharp shape could be measured thanks to the 0.08 meV broad Gaussian resolution function. Which enables a superior definition of the spectral shape. However, this performance imposes an overall shrinkage of the spanned frequency range, which does not include the high-frequency shoulder in the IXS spectrum. On the other hand, the resolution of the IXS measurement is too coarse to enable a proper definition of the quasielastic portion of the scattering profile.
In conclusion, we illustrated the main analytical steps leading to a derivation of the inelastic X-ray scattering, IXS signal, and demonstrated its direct link with the terahertz spectrum of atomic density fluctuations.
Since its development in the mid-1990s, high-resolution IXS has rapidly transitioned to its mature age, nowadays representing an essential tool to characterize the terahertz dynamics of liquid and amorphous materials. Historically, the mainstream scientific interest of the IXS community was mostly limited to simple fluids and glass-forming materials. In recent years, such a focus has gradually shifted towards nanostructured metamaterials and biological systems. Since the high complexity of these systems often challenges a firm understanding of the measurement outcome, a firm theoretical modeling of the IXS signal from these highly heterogeneous systems would be highly beneficial.
This work used resources of the National Synchrotron Light Source II, a U.S. Department of Energy Office of Science User Facility operated for the DOE Office of Science by Brookhaven National Laboratory under Contract No. DE-SC0012704.
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- For simple sample shapes, the treatment can be easily extended to the case of finite scattering angle by using simple trigonometry.